Wireless Ad Hoc and Sensor Networks

152 Wireless Ad Hoc and Sensor Networksand Sat (.) is a saturation function that satisfies the following:pSat () z = 0,ifz ≤ 0,This value of T fb is obtained from the round-trip delay time (RTT−RTT min), where RTT is the round-trip delay and RTT min is the minimumpropagation time obtained from the link bandwidth. The unknown disturbancevector acting on the system at the instant k is dk ( )∈R , which isnassumed to be bounded by a known constant |( dk)|≤ d M . Here, the disturbancevector dk ( ) can be an unexpected traffic burst/load or change inavailable bandwidth owing to the presence of a network fault. Thestate xk ( ) is a positive scalar if a single-switch or single-buffer scenario isconsidered, whereas it becomes a vector when multiple network switchesor multiple buffers are involved. The first step in the proposed approachis to estimate the buffer occupancy, using an estimate of the network trafficat the switch. For the sake of convenience, we will eliminate saturationby not allowing xk ( ) to saturate.The objective here is to construct a model to identify the traffic accumulationat the switch as:xk ˆ( + 1 ) = f ˆ( xk ( )) + Tuk ( ) , (4.2)nwhere the state of the model, xk ˆ( ) ∈R , is the buffer occupancy estimateat time instant k, with the nonlinear function f ˆ( x( k))being the trafficaccumulation estimate. Define the performance criterion in terms of bufferoccupancy estimation error aswhere the packet/cell losses, for a buffer size ofp= p,if z ≥ p,= z, otherwiseek ( ) = xk ( ) −xkˆ( ), (4.3)x d, are given byck ( ) = max( 0, ek ( ))(4.4)Equation 4.3 can be expressed using Equation 4.4 and Equation 4.5 as:ek ( + 1) = f̃( xk ( )) + dk ( ), (4.5)where ek ( + 1)and xk ( + 1)denote the error in buffer occupancy and thebuffer occupancy at the instant , respectively, and the traffic flowmodeling error is given by ̃f k + 1( x ( k )) = f ( x ( k )) − f ˆ( x ( k )) . Equation 4.5 relatesthe buffer occupancy estimation error with the traffic flow modeling error.